[Paper Review] 1--D Schrödinger operators with local interactions on a discrete set

Aleksey Kostenko, M. M. Malamud|arXiv (Cornell University)|Aug 25, 2009

Spectral Theory in Mathematical Physics20 citations

TL;DR

This paper investigates 1D Schrödinger operators with $δ$- and $δ'$-type point interactions on a discrete set $X = \{x_n\}_{n=1}^\infty$ where the infimum distance between points $d_* = 0$. Using boundary triplets and Weyl functions, it establishes a spectral correspondence between such operators and Jacobi matrices, deriving necessary and sufficient conditions for self-adjointness, lower semiboundedness, and discreteness in the $d_* = 0$ regime—revealing fundamental differences from the $d_* > 0$ case.

ABSTRACT

Spectral properties of 1-D Schrödinger operators $\mathrm{H}_{X,α}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in X}α_nδ(x-x_n)$ with local point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty$ are well studied when $d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. We consider $\mathrm{H}_{X,α}$ in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of $\mathrm{H}_{X,α}$ like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators $\mathrm{H}_{X,α}$ to be self-adjoint, lower-semibounded, and discrete in the case $d_*=0$. The operators with $δ'$-type interactions are investigated too. The obtained results demonstrate that in the case $d_*=0$, as distinguished from the case $d_*>0$, the spectral properties of the operators with $δ$ and $δ'$-type interactions are substantially different.

Motivation & Objective

• To analyze the spectral properties of 1D Schrödinger operators with local $δ$- and $δ'$-interactions on a discrete set $X$ when the infimum distance $d_* = 0$, a regime where standard methods fail.
• To extend the theory of self-adjoint extensions of symmetric operators to the case of dense point interactions using the framework of boundary triplets and Weyl functions.
• To establish a precise spectral correspondence between such Schrödinger operators and Jacobi matrices, enabling the transfer of known results on matrix spectra to the differential operator setting.
• To derive necessary and sufficient conditions for self-adjointness, lower semiboundedness, and discreteness of the operators in the $d_* = 0$ case, which are shown to differ significantly from the $d_* > 0$ case.

Proposed method

• The authors employ the theory of boundary triplets and Weyl functions to parametrize self-adjoint extensions of the minimal symmetric operator $\mathrm{H}_{\min}^*$ associated with the Schrödinger operator with point interactions.
• They construct a direct sum of boundary triplets for individual interaction points, showing that the resulting structure forms a boundary triplet for the full operator when the interaction strengths and geometry satisfy certain conditions.
• The spectral properties of the Schrödinger operator are linked to those of a Jacobi matrix via the Weyl function and Kreïn-type resolvent formula, enabling spectral transfer.
• The method relies on generalized boundary relations and the extension theory of symmetric operators, particularly for nonnegative operators with infinite deficiency indices.
• The analysis includes both $δ$- and $δ'$-interactions, with the latter treated via a modified boundary condition framework.
• The authors use the Berezanskii test and properties of Jacobi matrices to determine the number of deficiency indices and spectral type.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for self-adjointness of a 1D Schrödinger operator with $δ$-interactions on a discrete set $X$ when $d_* = 0$?
• RQ2 How do the spectral properties—such as lower semiboundedness and discreteness—of $δ$- and $δ'$-interaction operators differ when $d_* = 0$ compared to $d_* > 0$?
• RQ3 Can the spectral theory of Jacobi matrices be used to characterize the spectral type of Schrödinger operators with dense point interactions?
• RQ4 What role does the geometry of the interaction points (e.g., $d_n = 1/n$) play in determining the spectral properties of the resulting operator?
• RQ5 Under what conditions on the interaction strengths $\alpha_n$ and $\beta_n$ is the operator $\mathrm{H}_{X,\alpha}$ lower semibounded or discrete?

Key findings

• The operator $\mathrm{H}_{X,\alpha}$ is self-adjoint if and only if the associated Jacobi matrix $B_{X,\alpha,q_a}$ is self-adjoint, establishing a direct spectral correspondence.
• For $d_n = 1/n$ and $\alpha_n = -4n - 2$, the operator $\mathrm{H}_{X,\alpha,0}$ is self-adjoint, demonstrating that dense interactions can still yield self-adjoint realizations.
• When $a_0$ satisfies $\varepsilon_1(a_0) = 2$, the operator $\mathrm{H}_{X,\alpha,q_a}$ has deficiency indices $n_\pm = 1$, showing it is symmetric but not self-adjoint.
• The spectral type of the Schrödinger operator is determined by the behavior of the Jacobi matrix entries: if $\{b_n^{-1}\} \in \ell^1$ and $b_{n-1}b_{n+1} \leq b_n^2$, then $n_\pm = 1$ by Berezanskii's test.
• The study reveals that $\delta$- and $\delta'$-interactions exhibit fundamentally different spectral behavior when $d_* = 0$, unlike in the $d_* > 0$ case.
• The Weyl function of the Schrödinger operator is related to that of the Jacobi matrix via $M_n(z) = R_n^{-1}(\widetilde{M}_n(z) - Q_n)R_n^{-1}$, enabling spectral analysis through matrix theory.